Peakons and periodic cusp waves in a generalized Camassa–Holm equation

We study the peakons and the periodic cusp wave solutions of the following equation:ut+2kux−uxxt+auux=2uxuxx+uuxxxwith , which we will call the generalized Camassa–Holm equation, or simply the GCH equation, for when a=3 it was the so-called CH equation given by R. Camassa and D.D. Holm [Phys. Rev. Lett. 71 (11) (1993) 1661–1664]. They showed that the CH equation has a class of new solitary wave solutions called “peakons”. J.P. Boyd [Appl. Math. Comput. 81 (2–3) (1997) 173–187] studied another class of new periodic wave solutions called “coshoidal waves”. Using the bifurcation method of the phase plane, we first construct peakons and show that a=3 is the peakon bifurcation parameter value for the GCH equation. Then we construct some smooth periodic wave solutions, periodic cusp wave solutions, and oscillatory solitary wave solutions, and show their convergence when either the parameter a or the wave speed c varies. We also illustrate how to identify the existence of peakons and periodic cusp waves from the phase portraits. It seems that the GCH equation is a good example to understand the relationships among peakons, periodic cusp waves, oscillatory solitary waves and smooth periodic wave solutions.